Solve for r v = square root of (gm)/r

$v = \sqrt{\frac{g m}{r}}$

Step 1

Rewrite the equation as $\sqrt{\frac{g m}{r}} = v$ .

$\sqrt{\frac{g m}{r}} = v$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{g m}{r}}}^{2} = v^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{g m}{r}}$ as ${(\frac{g m}{r})}^{\frac{1}{2}}$ .

${({(\frac{g m}{r})}^{\frac{1}{2}})}^{2} = v^{2}$

Simplify the left side.

Simplify ${({(\frac{g m}{r})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{g m}{r})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{g m}{r})}^{\frac{1}{2} \cdot 2} = v^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{g m}{r})}^{\frac{1}{2} \cdot 2} = v^{2}$

Rewrite the expression.

${(\frac{g m}{r})}^{1} = v^{2}$

Simplify.

$\frac{g m}{r} = v^{2}$

Step 4

Solve for $r$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r, 1$

The LCM of one and any expression is the expression.

$r$

Multiply each term in $\frac{g m}{r} = v^{2}$ by $r$ to eliminate the fractions.

Multiply each term in $\frac{g m}{r} = v^{2}$ by $r$ .

$\frac{g m}{r} r = v^{2} r$

Simplify the left side.

Cancel the common factor of $r$ .

Cancel the common factor.

$\frac{g m}{r} r = v^{2} r$

Rewrite the expression.

$g m = v^{2} r$

Solve the equation.

Rewrite the equation as $v^{2} r = g m$ .

$v^{2} r = g m$

Divide each term in $v^{2} r = g m$ by $v^{2}$ and simplify.

Divide each term in $v^{2} r = g m$ by $v^{2}$ .

$\frac{v^{2} r}{v^{2}} = \frac{g m}{v^{2}}$

Simplify the left side.

Cancel the common factor of $v^{2}$ .

Cancel the common factor.

$\frac{v^{2} r}{v^{2}} = \frac{g m}{v^{2}}$

Divide $r$ by $1$ .

$r = \frac{g m}{v^{2}}$

Solve for r v = square root of (gm)/r

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